/ department of mathematics and computer science 3/11 PA3 PA2 PA1 axioms: equational theory. Defjnitions Equational theory / department of mathematics and computer science 4/11 A signature is a collection of symbols with an arity . Process Algebra (2IMF10) Syntax / department of mathematics and computer science Defjnitions PA4 / department of mathematics and computer science algebra of natural numbers Bas Luttik MF 6.072 s.p.luttik@tue.nl http://www.win.tue.nl/~luttik Lecture 1 2/11 Algebra of natural numbers Consider set of natural numbers N together with the familiar operations ▶ + : N × N → N ; and ▶ × : N × N → N . Let succ : N → N be the unary operation on N associating with every natural number its successor (i.e., succ ( n ) = n + 1 for all n ∈ N ). Then + , × , succ and 0 satisfy the following equations for all m , n ∈ N : n + 0 = n m + succ ( n ) = succ ( m + n ) n × 0 = 0 m × succ ( n ) = ( m × n ) + m Since every n ∈ N is obtained by the n -fold application of succ to 0 , this can be considered as a recursive defjnition of + and × . An equation is a formula of the form t = u , with t and u terms. An equational theory is a pair ( Σ , E ) consisting of a signature Σ and a set of Σ -equations E ; the equations in E are the axioms of the A signature Σ determines, together with a countably infjnite set of variables V , an inductively defjned set T ( Σ , V ) of terms (see book). The equational theory ( Σ 1 , E 1 ) of natural numbers has the following The signature Σ 1 of the theory natural numbers consists of ▶ a nullary function symbol (a.k.a. constant symbol ) 0 ; a ( x , 0) = x ▶ a unary function symbol s ; and a ( x , s ( y )) = s ( a ( x , y )) ▶ binary function symbols a and m . m ( x , 0) = 0 m ( x , s ( y )) = a ( m ( x , y ), x )
5/11 6/11 The quality of an equational theory / department of mathematics and computer science 8/11 on natural numbers: Model: linking logic and algebra / department of mathematics and computer science 7/11 / department of mathematics and computer science / department of mathematics and computer science Equational logic (example derivation) rules of equational logic: Equational logic Let T = ( Σ , E ) be an equational theory. We write T ⊢ t = u if there is a derivation of t = u using the following ( Axiom ) if t = u is an equation in E Consider the equational theory T 1 = ( Σ 1 , E 1 ) . t = u t = u t = u u = v Give a proof tree showing that T 1 ⊢ m ( s (0), s (0)) = a (0, s (0)) . ( Refm. ) ( Symm. ) ( Trans. ) t = t u = t t = v t = u ( Subst. ) σ : V → T ( Σ , V ) is a substitution t [ σ ] = u [ σ ] t 1 = u 1 · · · t n = u n ( Cont. ) f ∈ Σ of arity n f ( t 1 , . . . , t n ) = f ( u 1 , . . . , u n ) We defjne an interpretation ι of the function symbols in Σ 1 as functions Let T = ( Σ , E ) be an equational theory, and assume some fjxed interpretation of the function symbols in Σ in a Σ -algebra 1 A . 0 �→ 0 , A Σ -equation t = u is valid in A (notation: A | = t = u ) if ι α ( t ) = ι α ( u ) for all assignments α : V → N . s �→ succ , a �→ + , and ▶ T is sound for A : all equations that are derivable in T are valid in A . ▶ T is complete for A : all equations valid in A are derivable in T . m �→ × . ▶ T is ground-complete for A : all closed equations valid in A are Then, under an assignment α : V → N , every term in T ( Σ 1 , V ) denotes derivable in T . an element of N : ι α ( x ) = α ( x ) 1 A Σ -algebra is an algebra together with an interpretation of each n -ary ι α ( f ( t 1 , . . . , t n )) = ι ( f )( ι α ( t 1 ), . . . , ι α ( t n )) function symbol in Σ as an n -ary operation on A .
9/11 [Read the proof in the book. We will study a ground-completeness proof Exercise Completeness Theorem / department of mathematics and computer science 11/11 Proof. By the general soundness of the rules of equational logic (see Proposition 2.3.9 in the book) it suffjce to check that the axioms PA1–PA4 are valid in the context of process algebra.] 10/11 / department of mathematics and computer science / department of mathematics and computer science Theorem T 1 is sound for N T 1 is ground-complete for N Let N = ( N , +, × , succ , 0) , and let ι be the standard interpretation of a , m , s and 0 as + , × , succ and 0 in N . For all closed Σ 1 -terms p and q , if N | = p = q , then T 1 ⊢ p = q . For all Σ 1 -terms t and u , if T 1 ⊢ t = u , then N | = t = u . in N (Exercise!). Is T 1 complete for N ? No! For instance, N | = a ( x , y ) = a ( y , x ) , but T 1 ̸⊢ a ( x , y ) = a ( y , x ) . Argue that T 1 ̸⊢ a ( x , y ) = a ( y , x ) .
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